To find the vertical asymptote of the function \( f(x) = 2\ln(x) - \ln(d) \), we need to analyze when the function becomes undefined or approaches infinity.

### Steps:

1. The natural logarithm function \( \ln(x) \) is undefined for \( x \leq 0 \). Therefore, the domain of \( \ln(x) \) is \( x > 0 \).

2. The term \( 2\ln(x) \) implies that the function involves a logarithm of \( x \), so the expression becomes undefined if \( x \leq 0 \). This means \( x = 0 \) is a potential vertical asymptote.

3. For the function \( 2\ln(x) - \ln(d) \), the constant \( \ln(d) \) (where \( d \) is a positive constant) does not affect the position of the vertical asymptote because it just shifts the graph vertically. The key point is that the logarithmic term \( \ln(x) \) dominates.

### Conclusion:
The vertical asymptote occurs at **\( x = 0 \)**.

Would you like more details, or do you have any other questions?

Here are 5 related questions:
1. How do you find the horizontal asymptote of a logarithmic function?
2. What happens when \( d \) is negative in the given function?
3. How do transformations (shifts and stretches) affect logarithmic graphs?
4. What are the general properties of logarithmic functions?
5. Can logarithmic functions have more than one vertical asymptote?

**Tip:** The logarithmic function \( \ln(x) \) has a vertical asymptote at \( x = 0 \) for all real positive values of \( x \).

Discover how to find the vertical asymptote of the logarithmic function f(x) = 2ln(x) - ln(d). This solution explores key logarithmic concepts and properties, suitable for high school students in Grades 10-12.

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